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a2ee19a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 | import Mathlib
import OAI.Probability.Perceptron.Pressure.SphericalPatternDirection
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set
open scoped BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
lemma spherical_pattern_radial_ibp {n : ℕ}
(μ : Measure (NormalizedSpin (n+1))) [IsProbabilityMeasure μ]
(f : Jet3) {h : NormalizedSpin (n+1)→ℝ}
(hh : Measurable h) {A : ℝ} (hA : 0≤A) (hhA : ∀ x,|h x|≤A) :
let b := gaussianField (n+1) (fun i (x : NormalizedSpin (n+1)) => x.val i)
let H := fun g x => h x+f.f (b g x)
(∫ g,gibbsReplicaMean μ (H g) 1 (fun x => b g (x 0)*f.d1 (b g (x 0)))
∂Measure.pi (fun _ => gaussianReal 0 1))=
∫ g,gibbsReplicaMean μ (H g) 1 (fun x => f.d2 (b g (x 0))+f.d1 (b g (x 0))^2)-
gibbsReplicaMean μ (H g) 2 (fun x => spinOverlap (x 0) (x 1)*f.d1 (b g (x 0))*f.d1 (b g (x 1)))
∂Measure.pi (fun _ => gaussianReal 0 1) := by
dsimp only
let v := fun i (x : NormalizedSpin (n+1)) => x.val i
let w := fun i (x : Fin 1→NormalizedSpin (n+1)) => v i (x 0)
have hv : ∀ i,Measurable (v i) := fun i => by dsimp [v]; fun_prop
have hw : ∀ i,Measurable (w i) := fun i => by dsimp [w,v]; fun_prop
have hK (x : Fin 1→NormalizedSpin (n+1)) (y : NormalizedSpin (n+1)) :
(∑ i,w i x*v i y)=spinOverlap (x 0) y := by simp only [w,v,spinOverlap_sum]
have hb (g : Fin (n+1)→ℝ) (x : Fin 1→NormalizedSpin (n+1)) :
(∑ i,g i*w i x)=gaussianField (n+1) v g (x 0) := rfl
have he := nonlinear_pattern_direction_ibp μ (0:Fin 1) f hv hh hw hA
(by norm_num : (0:ℝ)≤1) (by norm_num : (0:ℝ)≤1) hhA
(fun i x => normalizedSpin_coord_bound x i) (fun i x => normalizedSpin_coord_bound (x 0) i)
dsimp only at he
simp_rw [hK,hb] at he
convert he using 1
· congr 1; funext g; congr 1; funext x; ring
· congr 1; funext g
simp only [Nat.cast_one,one_mul,Fin.sum_univ_one,spinOverlap_self,Fin.succ_zero_eq_one,mul_one]
congr 1
· congr 1; funext x; ring
· congr 1; funext x
rw [show spinOverlap (x 1) (x 0)=spinOverlap (x 0) (x 1) from real_inner_comm _ _]
ring
end SphericalPerceptronFreeEnergy
end
end OAI
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